Councillor Oldham said it was her aim to bring "harmony and peace" during her one-year term before Harrogate Borough Council is abolished next April.
She was sworn in at the Royal Hall and took over from Liberal Democrat councillor Trevor Chapman who served as mayor for the past 12 months.
Conservative councillor Robert Windass was also made deputy mayor.
Councillor Oldham - who represents the Washburn ward and works as an accountant and farmer - said: "Throughout the year there will be a need to take stock, but more importantly it is a matter of bringing people together.
"I'd like to thank councillor Trevor Chapman for all of his hard work in the last year.
"I naturally am looking forward to continuing with this civic tradition, albeit until Harrogate Borough Council unfortunately will be no more."
Also at tonight's meeting, re-elected council leader Richard Cooper appointed councillor Sam Gibbs as cabinet member for environment, waste reduction and recycling.
Councillor Gibbs succeeds councillor Andy Paraskos who was previously a member of the seven-person cabinet.
Meanwhile, councillor Rebecca Burnett, who previously oversaw the creation of Harrogate's Local Plan as cabinet member for planning, was made chair of the planning committee.
She replaces councillor John Mann who will remain a committee member.
Newly-elected mayor councillor Oldham also passed her previous title of chair of the licensing committee to councillor Ed Darling.
The role of the mayor is to chair full council meetings and represent the borough at ceremonies and events. They also raise money for charities and are required to put their political affiliations aside to be impartial.
Councillor Oldham has chosen the Yorkshire Air Ambulance as the first of her nominated charities.
Speaking at tonight's meeting, councillor Cooper said councillor Oldham will be an "outstanding mayor for all of our borough."
He said: "During this year she will meet hundreds of community groups, businesses and individuals. They will see in her what many of us have seen since we met her in 2017.
"She will see us out with flair, fun and dedication."
Councillor Cooper - who will step down as the longest-serving leader of the borough council next April - also said the authority would continue pushing ahead with major projects during its final year before the new North Yorkshire Council takes over.
He said: "I pledge to you - on behalf of the cabinet - that this will not be a year of winding down. It will be a year of handover.
"As part of that handover, we continue to have a string of exciting initiatives and projects to take forward.
"And in the cabinet I have appointed, we have a team that can do a great job for the people of the borough."
By Jacob Webster, Local Democracy Reporter